Calculus Examples

$lim_{n \to 8} \frac{1}{n^{\frac{1}{\ln (n)}}}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $8$ .

$\frac{lim_{n \to 8} 1}{lim_{n \to 8} n^{\frac{1}{\ln (n)}}}$

Step 1.2

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$\frac{1}{lim_{n \to 8} n^{\frac{1}{\ln (n)}}}$

Step 2

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 2.1

Rewrite $n^{\frac{1}{\ln (n)}}$ as $e^{\ln (n^{\frac{1}{\ln (n)}})}$ .

$\frac{1}{lim_{n \to 8} e^{\ln (n^{\frac{1}{\ln (n)}})}}$

Step 2.2

Expand $\ln (n^{\frac{1}{\ln (n)}})$ by moving $\frac{1}{\ln (n)}$ outside the logarithm.

$\frac{1}{lim_{n \to 8} e^{\frac{1}{\ln (n)} \ln (n)}}$

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Move the limit into the exponent.

$\frac{1}{e^{lim_{n \to 8} \frac{1}{\ln (n)} \ln (n)}}$

Step 3.2

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $8$ .

$\frac{1}{e^{lim_{n \to 8} \frac{1}{\ln (n)} \cdot lim_{n \to 8} \ln (n)}}$

Step 3.3

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $8$ .

$\frac{1}{e^{\frac{lim_{n \to 8} 1}{lim_{n \to 8} \ln (n)} \cdot lim_{n \to 8} \ln (n)}}$

Step 3.4

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$\frac{1}{e^{\frac{1}{lim_{n \to 8} \ln (n)} \cdot lim_{n \to 8} \ln (n)}}$

Step 3.5

Move the limit inside the logarithm.

$\frac{1}{e^{\frac{1}{\ln (lim_{n \to 8} n)} \cdot lim_{n \to 8} \ln (n)}}$

Step 3.6

Move the limit inside the logarithm.

$\frac{1}{e^{\frac{1}{\ln (lim_{n \to 8} n)} \cdot \ln (lim_{n \to 8} n)}}$

Step 4

Evaluate the limits by plugging in $8$ for all occurrences of $n$ .

Tap for more steps...

Step 4.1

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{1}{e^{\frac{1}{\ln (8)} \cdot \ln (lim_{n \to 8} n)}}$

Step 4.2

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{1}{e^{\frac{1}{\ln (8)} \cdot \ln (8)}}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Cancel the common factor of $\ln (8)$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$\frac{1}{e^{\frac{1}{\ln (8)} \cdot \ln (8)}}$

Step 5.1.2

Rewrite the expression.

$\frac{1}{e^{1}}$

Step 5.2

Simplify.

$\frac{1}{e}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e}$

Decimal Form:

$0.36787944 \dots$